Induced generalized dual hesitant fuzzy Shapley hybrid operators and their application in multi-attributes decision making

Abstract

In this study, two induced generalized dual hesitant fuzzy hybrid operators called the induced generalized dual hesitant fuzzy Shapley hybrid weighted averaging (IG-DHFSHWA) operator and the induced generalized dual hesitant fuzzy Shapley hybrid geometric mean (IG-DHFSHGM) operator are defined. These operators not only globally consider the importance of the elements and their ordered positions, but also overall reflect their correlations. Furthermore, when the weight information about attributes and ordered positions is partly known, using similarity measure analysis (SMA) method and the Shapley function models for the optimal fuzzy measures on an attribute set and on an ordered set are respectively established. Then, an approach to multi-attributes decision making with incomplete weight information and interactive conditions under dual-hesitant fuzzy environment is developed. Finally, a practical example for green supplier evaluation problem with dual hesitant fuzzy information is employed to verify the developed approach and to demonstrate its practically and effectiveness.

Keywords

Multi-attribute decision making dual hesitant fuzzy sets the similarity measure analysis method Shapley functions

1 Introduction

Since Zadeh [1] introduced the concept of fuzzy set (FSs), many extensions of the theory have been proposed, such as intuitionistic fuzzy set [2], interval-valued intuitionistic fuzzy set [3], interval-valued fuzzy set [4], linguistic fuzzy set [5], type-2 fuzzy set [6, 7], type-n fuzzy set [6], and fuzzy multiset [8, 9]. In some cases, we often encounter fuzzy situations in which it is difficult to determine the membership of an element to a set due to doubts between a few different values. For example, when a customer decide which car to buy with respect to a given attributes, who may invite many experts to estimate the performance of the car. Some experts regard 0.4 as the membership degree of performance, some regard 0.8 as the membership degree of performance, other regard 0.5 as the membership degree of performance. The three experts cannot compromise and change their evaluation. In this case, a simpler method is that the membership degree of performance should be composed of the three possible values. To solve this kind of problems, Torra and Narukawa [10] and Torra [8] introduced the concept of hesitant fuzzy set (HFS), which permits the membership having a set of possible values. For the above example, it can be represented by a hesitant fuzzy set {0.4, 0.5, 0.8}. More and more multiple attribute decision making theories and methods under hesitant fuzzy environment have been studied since its appearance[11, 12]. Xia and Xu [11] defined some operational laws and gave an intensive study on hesitant fuzzy information aggregation techniques and their application in decision making. Xu and Xia [13] gave a detailed study on distance and similarity measures for hesitant fuzzy set (HFS) and proposed an approach based on distance measures for multi-attribut decision making (MADM) problems. Motivated by the idea of the prioritized aggregation operator, Wei [12] proposed some prioritized aggregation operators for hesitant fuzzy information, and developed some models for hesitant fuzzy MADM problems in which the attributes are in different priority levels. Based on Choquet integral, Wei et al. [14] developed some hesitant fuzzy Choquet integral aggregation operators and applied them to multiple attribute decision making. Based on Shapley function, Wei et al. [15] compared VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje) method with TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) method for multiple attribute decision making problems under hesitant fuzzy environment. Recently, hesitant fuzzy set has been widely extended. Interval-valued hesitant fuzzy set (IVHFS) was introduced by Chen et al. [16, 17] and Wei et al. [18], who generalized the concept of Torra’s HFS [10] in which each element in IVHFS is expressed by an ordered pair, and each ordered pair is characterized by several membership and non-membership degrees with several possible interval values. Motivated fuzzy linguistic decision analysis theory [19], Rodríguez et al. [20] proposed the concept of hesitant fuzzy linguistic term set (HFLTS). In particular, considering that people preferences information may be often expressed several possible membership degrees with linguistic variables and uncertain linguistic variables, Lin et al. [21] proposed the concepts of hesitant fuzzy linguistic set (HFLS) and hesitant fuzzy uncertain linguistic set (HFULS). Combing linguistic fuzzy set and hesitant fuzzy set, Meng et al. [22] defined linguistic hesitant fuzzy sets (LHFSs), gave its operational laws and proposed generalized linguistic hesitant fuzzy hybrid weighted averaging (GLHFHWA) operator and generalized linguistic hesitant fuzzy hybrid geometric mean (GLHFHGM) operator.

Furthermore, in order to accommodate decision needs to people daily life, Zhu et al. [23] extended hesitant fuzzy set and proposed dual hesitant fuzzy set (DHFS), which is comprehensive set encompassing fuzzy set [24], intuitionistic fuzzy set [2], hesitant fuzzy set [8, 10] and fuzzy multiset [25], and whose membership and non-membership are represented by a set of several possible values. Owing to DHFSs desirable characteristics, it appears to be a more flexible method than most of the existing other fuzzy set, receiving much more attention from researchers now. Wang et al. [26] developed some dual hesitant fuzzy aggregation operators, such as dual hesitant fuzzy weighted average (DHFWA) operator, dual hesitant fuzzy weighted geometric (DHFWG) operator, dual hesitant fuzzy ordered weighted average (DHFOWA) operator, dual hesitant fuzzy ordered weighted geometric (DHFOWG) operator. Yang et al. [27] defined several new dual hesitant fuzzy aggregation operators, such as dual hesitant fuzzy linguistic weighted average (DHFLWA) operator, dual hesitant fuzzy linguistic weighted geometric (DHFLWG) operator, dual hesitant fuzzy linguistic ordered weighted average (DHFLOWA) operator, dual hesitant fuzzy linguistic ordered weighted geometric (DHFLOWG) operator. However, all above dual hesitant hybrid aggregation operators are based on the assumption that the discussed elements in a set are independent, i.e., they only consider the addition of importance of individual elements. For real decision making problems, in many practical situations, the elements in a set are usually correlative [28 –34].

To cope with situation where the elements in a set are correlative, Wang and Ju [35] presented generalized dual hesitant fuzzy Choquet ordered aggregation (GDHFCOA) operator. Ju et al. [36] used Choquet integral to propose some dual hesitant fuzzy aggregation operators based on fuzzy measures, such as the dual hesitant fuzzy Choquet ordered average (DHFCOA) operator, dual hesitant fuzzy Choquet ordered geometric (DHFCOG) operator, generalized dual hesitant fuzzy Choquet ordered geometric (GDHFCGM) operator. All these operators are based on the assumption that the fuzzy measure on an attributes set is completely known. However, in real decision making, due to time pressure, lack of knowledge or the expert’s limited expertise about the problem domain, the information about attribute weights is often incompletely known. As Meng et al. noted [37 –40], the Choquet integral only reflects the interaction between two adjacent coalitions. Furthermore, the Choquet integral can either give the importance of elements or that of ordered positions, but it cannot both consider these twoaspects.

To cope with these issues, this study presents two new induced generalized dual hesitant fuzzy hybrid Shapley operators, which are named as the induced generalized dual hesitant fuzzy Shapley hybrid weighted averaging (IG-DHFSHWA) operator and the induced generalized dual hesitant fuzzy Shapley hybrid geometric mean (IG-DHFSHGM) operator, respectively. These operators not only globally consider the importance of elements and their ordered positions, but also overall reflect the interaction among them and among their ordered positions. Furthermore, when the weight vectors about attribute and ordered position are partly known or incompletely known, models for the optimal fuzzy measures on the attribute set and the ordered position set are established, respectively, by which the weight vectors on them can be obtained. As a series of extension, a method to dual hesitant fuzzy multi-attribute decision making with incomplete weight information and interaction conditions is developed. In order to consider these, the remainder of this paper is organized as follows:

In Section 2, some basic conceptions and notations about dual hesitant fuzzy sets (DHFSs), fuzzy measure, Shapley function and the Choquet integral are reviewed, which are used in the rest parts of the paper. In Section 3, the induced generalized dual hesitant fuzzy Shapley hybrid weighted averaging (IG-DHFSHWA) operator and the induced generalized dual hesitant fuzzy Shapley hybrid geometric mean (IG-DHFSHGM) operator are defined, and some important cases are examined. In Section 4, based on the similarity analysis method and the Shapley function, models for the optimal fuzzy measures on the attribute set and on the ordered set are constructed to determine weight vectors on attribute set and ordered set. In Section 5, an approach to dual hesitant fuzzy multi-attribute decision making with incomplete information weights and interaction conditions is developed. In Section 6, a numerical example has been given to illustrate the developed approach. Conclusions and future research topics are presented in the final section.

2 Preliminaries

For the convenience of analysis, some basic concepts are reviewed to facilitate future discussions.

2.1 Hesitant fuzzy set and dual hesitant fuzzy set

Torra and Narukawa [10] and Torra [8] extended the fuzzy set to hesitant fuzzy set (HFS), which permits the membership degree of an element to a given set having a set of possible values.

Definition 1. [8, 10] Let X be a fixed set, then a HFS on X is in terms of a function that when applied to X returns a subset of [1], which can be represented as the following mathematical symbol: $E = {< x, h_{E} (x) > | x \in X}$ (1) where h_E (x) is a set of some values in [1], denoting the possible membership degrees of the element x ∈ X to the set E. For convenience, Xia and Xu [11] called h_E (x) a hesitant fuzzy element (HFE) and E the set of all hesitant fuzzy elements (HFEs).

Definition 2. [11] Let h, h₁ and h₂ be any three HFEs and γ > 0, then some operational laws about HFEs are defined as follows: $(1) h^{λ} = ⋃_{r \in h} {r^{λ}};$ (2) $(2) λ h = ⋃_{r \in h} {1 - (1 - r)^{λ}};$ (3) $(3) h_{1} \oplus h_{2} = ⋃_{r_{1} \in h_{1}, r_{2} \in h_{2}} {r_{1} + r_{2} - r_{1} r_{2}};$ (4) $(4) h_{1} \otimes h_{2} = ⋃_{r_{1} \in h_{1}, r_{2} \in h_{2}} {r_{1} r_{2}} .$ (5)

However, the HFS only considers the membership degree of an element to a given set without considering the non-membership degree. In order to overcome this limitation, Zhu et al. [23] developed another generalization of fuzzy set called dual hesitant fuzzy set.

Definition 3. [23] Let X be a fixed set, then a dual hesitant fuzzy set (DHFS) D on X is defined as

$D = {< x, h (x), g (x) > | x \in X},$ (6) where h (x) and g (x) are two set of some values in [1], denoting the possible membership degrees and non-membership degrees of the element x ∈ X to the set D, respectively, with the conditions: r, η ∈ [0, 1] and 0 ≤ r⁺+ η⁺≤ 1, where r ∈ h (x), η ∈ g (x), r⁺∈ h⁺(x) = ⋃ _r∈h(x) max {r} and η⁺∈ g⁺(x) = ⋃ _η∈g(x) max {η} for all x ∈ X. For convenience, we call the pair d (x) = {h (x) , g (x)} a dual hesitant fuzzy element (DHFE) denoted by d = {h, g} and Ω the set of all DHFEs.

Obviously, if there is only one element in both h (x) and g (x), the DHFE reduces to an intuitionistic fuzzy number [2, 3].

Definition 4. [23] Let d = {h, g}, d₁ = {h₁, g₁} and d₂ = {h₂, g₂} be any three DHFEs, then the operational laws of DHFEs are defined as follows:

$(1) d^{β} = ⋃_{r \in h, η \in g} {{{r}^{β}}, {1 - (1 - η)^{β}}}, β > 0;$ (7)

$(2) β d = ⋃_{r \in h, η \in g} {{{1 - (1 - r)^{β}}, {η}^{β}}}, β > 0;$ (8) $\begin{matrix} (3) d_{1} \oplus d_{2} = ⋃_{r_{1} \in h_{1}, r_{2} \in h_{2}, η_{1} \in g_{1}, η_{2} \in g_{2}} \\ {{r_{1} + r_{2} - r_{1} r_{2}}, {η_{1} η_{2}}}; \end{matrix}$ (9)

$\begin{matrix} (4) d_{1} \otimes d_{2} = ⋃_{r_{1} \in h_{1}, r_{2} \in h_{2}, η_{1} \in g_{1}, η_{2} \in g_{2}} \\ {{r_{1} r_{2}}, {η_{1} + η_{2} - η_{1} η_{2}}} . \end{matrix}$ (10)

Obviously, the above operational results are still DHFEs.

Definition 5. [23] Let d_i = (h_i, g_i) (i = 1, 2) be any two DHFEs, then the score function of d_i is $S (d_{i}) = \frac{1}{# h_{i}} \sum_{r_{i} \in h_{i}} r_{i} - \frac{1}{# g_{i}} \sum_{η_{i} \in g_{i}} η_{i}$ (i = 1, 2) and the accuracy function of d_i is $P (d_{i}) = \frac{1}{# h_{i}} \sum_{r_{i} \in h_{i}} r_{i} + \frac{1}{# g_{i}} \sum_{η_{i} \in g_{i}} η_{i}$ (i = 1, 2), where # h_i and # g_i are the numbers of values in h_i and g_i, respectively, then

(1) if S (d₁) > S (d₂), then d₁ is superior to d₂, denoted by d₁ > d₂;

(2) if S (d₁) = S (d₂), then

(a) if P (d₁) > P (d₂), then d₁ is superior to d₂, denoted by d₁ > d₂;

(b) if P (d₁) = P (d₂), then d₁ is equivalent to d₂, denoted by d₁ = d₂.

Definition 6. [36, 41] Let d_i (i = 1, 2, …, n) be a collection of DHFEs in D. w = (w_1,w₂, … w_n)^T be the weight vector on {d_i} (i = 1, 2, …, n) with $\sum_{i = 1}^{n} w_{i} = 1$ , and ω = (ω_1,ω₂, … ω_n)^T be the associated weight vector on ordered set N ={ 1, 2, … n } with ω_i ∈ [0, 1] and $\sum_{i = 1}^{n} ω_{i} = 1$ .

(1) The generalized dual hesitant fuzzy hybrid averaging (GDHFHA) operator is defined by $GDHFHA (d_{1}, d_{2}, \dots, d_{n}) = {(\oplus_{j = 1}^{n} ω_{j} {\dot{d}}_{σ (j)}^{γ})}^{\frac{1}{γ}} = ⋃_{r_{σ (j)} \in {\dot{h}}_{σ (j)}} {\begin{matrix} {{(1 - \prod_{j = 1}^{n} {(1 - r_{σ (j)}^{γ})}^{ω_{j}})}^{1 / γ}}, \\ {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - η_{σ (j)})}^{γ})}^{ω_{j}})}^{1 / γ}} \end{matrix}},$

where γ > 0, σ (j) is a permutation on the weighted DHFEs nw_id_i (i = 1, 2, …, n) with ${\dot{d}}_{σ (j)} = {nw}_{σ (j)} d_{σ (j)}$ being the jth largest value of nw_id_i (i = 1, 2, …, n), and n is the balancing coefficient.

(2) The generalized dual hesitant fuzzy hybrid geometric (GDHFHG) operator is defined by $GDHFHG (d_{1}, d_{2}, \dots, d_{n}) = \frac{1}{γ} (\otimes_{j = 1}^{n} {(γ {\dot{d}}_{σ (j)})}^{ω_{j}}) = ⋃_{r_{σ (j)} \in {\dot{h}}_{σ (j)}} {\begin{matrix} {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - r_{σ (j)})}^{γ})}^{ω_{j}})}^{1 / γ}}, \\ {{(1 - \prod_{j = 1}^{n} {(1 - η_{σ (j)}^{γ})}^{ω_{j}})}^{1 / γ}} \end{matrix}},$

where γ > 0, σ (j) is a permutation on the weighted DHFEs $d_{i}^{{nw}_{i}}$ (i = 1, 2, …, n) with ${\dot{d}}_{σ (j)} = d_{σ (j)}^{{nw}_{σ (j)}}$ being the jth largest value of $d_{i}^{{nw}_{i}}$ (i = 1, 2, …, n), and n is the balancing coefficient.

Although the GDHFHA and GDHFHG operators both consider the importance of elements and their ordered positions, they are based on the assumption that the elements in a set are independent. In the practical decision-making problems, this assumption is too strong to match decision behaviors in the real world, and the independence axiom generally cannot be satisfied, so it is unsuitable to aggregate their values using additive measures. In 1974, Sugeno [42] introduced the following concept of fuzzy measure (non-additive measure).

2.2 Fuzzy measure and Shapley function

Definition 7. [42] Let N = {1, 2, …, n} be a finite set and P (N) be the power set of N. The braces are often omitted for singletons, e.g., writing N ∖ i, T, N ∖ A instead of S ∖ { i } , {T } , N∖ { A }. Moreover, the cardinality of any subsets T ∈ P (N) will be denoted whenever possible by the corresponding lower case letters t.

The set function μ : P (N) → [0, 1] is called a fuzzy measure satisfying the following axioms:

(1) μ (φ) = 0, μ (N) = 1;

(2) If A, B ∈ P (N) and A ⊆ B then μ (A) ≤ μ (B);

To avoid the problems with computational complexity and practical estimations, a special kind of fuzzy measure named the λ-fuzzy measure Sugeno [42], which is expressed by the following form: $μ (C \cup D) = μ (C) + μ (D) + λ μ (C) μ (D),$ where λ > -1, and for all C, D ∈ P (N) with C ∩ D = Ø .

(1) If λ = 0, C and D are considered to be without interaction which is called an additive measure.

(2) If λ ≠ 0, and λ > 0, C and D exhibit a positive synergetic interaction between them, which is called a super-additive measure.

(3) If λ < 0, C and D exhibit a negative synergetic interaction between them, which is called a sub-additive measure.

To reflect the interactive characteristics between elements, Ju et al. [36] defined the following generalized dual hesitant fuzzy Choquet integral operators.

Definition 8. [36, 41] Let d_j (j = 1, 2, …, n) be a collection of DHFEs, and μ be a fuzzy measure on N = {1, 2, …, n}.

(1) The generalized dual hesitant fuzzy Choquet ordered average (GDHFCOA) operator is defined as follows:

$\begin{matrix} {GDHFCOA}_{μ} {d_{1}, d_{2}, \dots, d_{n}} \\ = \oplus_{j = 1}^{n} {((μ (A_{σ (i)}) - μ (A_{σ (i + 1)})) d_{σ (i)}^{γ})}^{1 / γ}, \end{matrix}$ (11)

(2) The generalized dual hesitant fuzzy Choquet ordered geometric (DHFCOG) operator is defined as follows:

$\begin{matrix} {GDHFCOG}_{μ} {d_{1}, d_{2}, \dots, d_{n}} \\ = \frac{1}{γ} (\otimes_{i = 1}^{n} {(γ d_{σ (i)})}^{(μ (A_{σ (i)}) - μ (A_{σ (i + 1)}))}) . \end{matrix}$ (12) where γ > 0, σ (.) indicates a permutation on N such that σ (i) indicates a permutation on N such that d_σ(1) ≤ d_σ(2) ≤ ⋯ ≤ d_σ(n), and A_σ(i) ={ i, …, n } with A_σ(n+1) = φ.

From Definition 8, one can observe that both GDHFCOA and GDHFCOG operators reflect the importance of their ordered positions and consider their correlations. However, they neither reflect importance of elements nor consider their correlations.

The Shapley function is one of the most important payoff indices, which has been deeply researched in game theory [15 , 43]. In order to measure the overall influence of each coalition other than each player in a game. The concept of the generalized value was proposed by Marichal [44] and Meng et al. [37] applied the generalized Shapley index to decision making. When it is calculated with respect to the fuzzy measure μ on finite set N = {1, 2, …, n}, it was denoted by

$\begin{matrix} ρ_{K}^{Sha} (μ, N) \\ = \sum_{O \subseteq N ∖ K} \frac{(n - k - o)! o!}{(n - k + 1)!} \\ (μ (K \cup O) - μ (O)), \forall K \subseteq N . \end{matrix}$ (13) where k, o and n respectively denote the cardinalities of KO and N.

From Equation (13) if K = { i } , then

$\begin{matrix} ρ_{i}^{Sha} (μ, N) \\ = \sum_{K \subseteq N ∖ i} \frac{(n - k - 1)! k!}{n!} \\ (μ (K \cup i) - μ (K)), \forall i \in N . \end{matrix}$ (14)

Remark 1. In fact, Equation (13) is an expected value of the overall interaction between the coalition K and every coalition in N ∖ K. As a special case, Equation (14) gives an expected value of the overall interaction between the element i and every coalition in N ∖ i.

From Equation (14), one can observe that $p_{n}^{k} = \frac{(n - k - 1)! k!}{n!}$ is a probability weight with $\sum_{K \subseteq N ∖ i} p_{n}^{k} = 1$ and $p_{n}^{k} > 0$ for each K ⊆ N. The Shapley function is actually an expectation value of the marginal contribution between the element i and any coalition K ⊆ N ∖ i. From the definition of fuzzy measures, it is not difficult to know that $ρ_{i}^{Sha} (μ, N) \geq 0$ for any i ∈ N and $\sum_{i = 1}^{n} ρ_{i}^{Sha} (μ, N) = 1$ . It means that ${ρ_{i}^{Sha} (μ, N)}_{i \in N}$ is a weight vector.

It is worth pointing out that if there is not any interaction between attributes, then their Shapley value can be seen as an extension of additive weights. In the multi-attributes decision making, if we consider the interactions between attributes, then the Shapley function may be a good choice.

Aggregation operators, as an important research topic in decision-making theory, one of the most importance aggregation operators is the ordered weighted averaging (OWA) operator [25]. This aggregation operator provides a parameterized family of aggregation operators between the minimum and the maximum [45]. Merigó et al. [46] extended OWA operator and developed some fuzzy generalized hybrid aggregation operators, such as the fuzzy generalized hybrid averaging (FGHA) and the fuzzy induced generalized hybrid averaging (FIGHA) operator.

The induced generalized aggregation operator is an important kind of aggregation operators that has been researched by many scholars, such as the induced ordered weighted average (IOWA) operator [47], induced generalized ordered weighted averaging (IGOWA) operator [48], the fuzzy induced generalized ordered weighted averaging (FIGOWA) and the fuzzy induced quasi-arithmetic OWA (Quasi- FIOWA) operator [49], the induced generalized intuitionistic fuzzy ordered weighted average (IGIFOWA) operator [50, 51], the induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging operator (IG-IVIFHSA) [29], the fuzzy induced ordered weighted averaging –weighted averaging (FIOWAWA) operator [46]. This operator allows analysis of a wide range of scenarios from the minimum (pessimistic) to the maximum (optimistic) and selects the one in closest accordance with our interests.

3 A new induced generalized dual hesitant fuzzy operators

To overcome above mentioned dual hesitant fuzzy aggregation operators only consider the importance of their ordered positions and reflect their correlations, a pair of new dual hesitant fuzzy operators called the induced generalized dual hesitant fuzzy Shapley hybrid weighted averaging (IG-DHFSHWA)operator and the induced generalized dual hesitant fuzzy Shapley hybrid geometric mean(IG-DHFSHGM) operator are introduced in this section, which overall consider the importance of their ordered positions and elements as well as the interaction among elements and their combinations.

Definition 9. An IG-DHFSHWA operator of dimension n is a mapping IG-DHFSHWA: Dⁿ→ D on the set of second arguments of two tuples <u₁, d₁> , < u₂, d₂ > , … , < u_n, d_n > with a set of order-inducing variables u_i (i = 1, 2, … n) and a parameter γ such that γ ∈ (0, + ∞), denoted by

$\begin{matrix} IG - DHFSHWA \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) \\ = {(\frac{\oplus_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L) d_{σ (j)}^{γ}}{\sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L)})}^{1 / γ}, \end{matrix}$ (15) where σ (i) indicates a permutation on u_i (i = 1, 2, … n) with u_σ(j) being the jth largest value u_i (i = 1, 2, …, n), ρ_j (μ, N) is the Shapley value with respect to (w.r.t.) the associated fuzzy measure μ on N ={ 1, 2, …, n } for the jth ordered position and ρ_{d
_σ(j)} (τ, L) is the Shapley value w.r.t. the fuzzy measure τ on L ={ d_j } (j = 1, 2, …, n) for d_j (j = 1, 2, …, n).

Remark 2. If μ and τ are both additive measure, then the IG-DHFSHWA operator degenerates to the induced generalized dual hesitant fuzzy hybrid weighted averaging (IG-DHFHWA) operator

$\begin{matrix} IG - DHFHWA \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) \\ = {(\frac{\oplus_{j = 1}^{n} w_{j} ω_{d_{σ (j)}}^{γ} d_{σ (j)}^{γ}}{\sum_{j = 1}^{n} ω_{d_{σ (j)}}^{γ} d_{σ (j)}^{γ}})}^{1 / γ}, \end{matrix}$ (16) where w_j = μ (j) and ω_{d
_j} = τ (d_j) for j = 1, 2, …, n.

Remark 3. When each DHFE d_i = (h_i, g_i) degenerates to be an intuitionistic fuzzy number, namely, there is only one element in both h_i and g_i for all i = 1, 2, …, n. Then IG-DHFSHWA operator degenerates to induced generally intuitionistic fuzzy hybrid Shapley averaging (IG-IFSHA) operator [29], denoted by

$\begin{matrix} IG - IFSHA \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) \\ = {(\frac{\oplus_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L) d_{σ (j)}^{γ}}{\sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L)})}^{1 / γ}, \end{matrix}$ (17) where ρ_{d
_σ(j)} (τ, L) d_σ(j) is the jth largest value of the Shapley weighted intuitionistic fuzzy ρ_{d
_i} (τ, L) d_i (i = 1, 2, …, n).

Furthermore, when each d_i = (h_i, g_i) degenerates to be fuzzy set, namely, there is only one element in h_i, and g_i =Ø for all i = 1, 2, …, n. then we get the induced generalized fuzzy hybrid Shapley averaging (IG-FHSA) operator

$\begin{matrix} IG - FSHA \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) \\ = {(\frac{\sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L) d_{σ (j)}^{γ}}{\sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L)})}^{1 / γ} . \end{matrix}$ (18)

Remark 4. If u_i = u_j for all i, j = 1, 2, …, n with i ≠ j. then the IG-DHFSHWA operator degenerates to the generalized dual hesitant fuzzy Shapley hybrid OWA(G-DHFSHOWA) operator

$\begin{matrix} G - DHFSHOWA (d_{1}, d_{2}, \dots, d_{n}) \\ = {(\frac{\oplus_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L) d_{σ (j)}^{γ}}{\sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L)})}^{1 / γ}, \end{matrix}$ (19) where ρ_{d
_σ(j)} (τ, L) d_σ(j) is the jth largest value of the Shapley weighted DHFEs ρ_{d
_i} (τ, L) d_i (i = 1, 2, …, n).

Remark 5. If γ = 1, then the IG-DHFSHWA operator degenerates to the induced dual hesitant fuzzy Shapley hybrid weighted averaging (I-DHFSHWA) operator

$\begin{matrix} I - DHFSHA \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) \\ = \frac{\oplus_{j = 1}^{n} ρ_{j} (μ, N) ρ_{σ (j)} (τ, L) d_{σ (j)}}{\sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{σ (j)} (τ, L)} . \end{matrix}$ (20)

Remark 6. If γ = 2, then the IG-DHFSHWA operator degenerates to the induced dual hesitant fuzzy Shapley hybrid quadratic weighted averaging(I-DHFSHQWA) operator

$\begin{matrix} I - DHFSHQWA \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) \\ = {(\frac{\oplus_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{2} (τ, L) d_{σ (j)}^{2}}{\sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{2} (τ, L)})}^{1 / 2} . \end{matrix}$ (21)

Remark 7. If τ (d_i) = 1/ - n for each i = 1, 2, …, n, then IG-DHFSHWA operator degenerates to the induced generalized dual hesitant fuzzy Shapley OWA (IG-DHFSOWA) operator

$\begin{matrix} I - DHFSOWA \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) \\ = {(\oplus_{j = 1}^{n} ρ_{j} (μ, N) d_{σ (j)}^{γ})}^{1 / γ} . \end{matrix}$ (22)

Furthermore If γ → 0⁺, then the IG-DHFSHWA operator degenerates to the induced dual hesitant fuzzy Shapley ordered geometric mean (I-DHFSOGM) operator

$\begin{matrix} I - DHFSOGM \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) \\ = \otimes_{j = 1}^{n} d_{σ (j)}^{ρ_{j} (μ, N)} . \end{matrix}$ (23)

Remark 8. If μ (j) = 1/ - n for each j = 1, 2, … n, then the IG-DHFSHWA operator degenerates to the induced generalized dual hesitant fuzzy Shapley weighted averaging (IG-DHFSWA) operator

$\begin{matrix} IG - DHFSWA \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) \\ = {(\frac{\oplus_{j = 1}^{n} ρ_{d_{σ (j)}}^{γ} (τ, L) d_{σ (j)}^{γ}}{\sum_{j = 1}^{n} ρ_{d_{σ (j)}}^{γ} (τ, L)})}^{1 / γ} . \end{matrix}$ (24)

Theorem 1.Let <u₁, d₁ (h₁, g₁)> , …, < u_n, d_n (h_n, g_n) > be a set of two tuples with a set of order-inducing variablesu_i (i = 1, 2, …, n) andd_i (i = 1, 2, …, n) being a collection of DHFEs in D, letμbe a fuzzy measure on the ordered setN = {1, 2, …, n}, and letτbe the associated fuzzy measure onL = { d_i } _{i∈{1,2,…,n}}. Then, their aggregated value using the IG-DHFSHWA operator is also a DHFE, denoted by

$\begin{matrix} IG - DHFSHWA (\begin{matrix} < u_{1}, d_{1} (h_{1}, g_{1}) >, \\ < u_{2}, d_{2} (h_{2}, g_{2}) >, \\ \dots, < u_{n}, d_{n} (h_{n}, g_{n}) > \end{matrix}) \\ = ⋃_{r_{σ (1)} \in h_{σ (1)}, η_{σ (1)} \in g_{σ (1),} r_{σ (2)} \in h_{σ (2),} η_{σ (2)} \in g_{σ (2), \dots,} r_{σ (n)} \in h_{σ (n)}, η_{σ (n)} \in g_{σ (n)}} \\ {{{(1 - \prod_{j = 1}^{n} (1 - r_{σ (j)}^{γ})^{ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L) / - \sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L)})}^{1 / γ}} \\ {1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - η_{σ (j)})}^{γ})}^{ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L) / - \sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}}^{γ} (τ, L)})}^{1 / γ}}}, \end{matrix}$ (25) where the notations as shown in Definition 9.

In a similar way to the IG-DHFSHWA operator, we can define the following induced generalized dual hesitant fuzzy Shapley hybrid geometric mean IG— DHFSHGM) operator.

Definition 10. An IG-DHFSHGM operator of dimension n is a mapping IG-DHFSHGM: Dⁿ→ D on the set of second arguments of two tuples <u₁, d₁> , < u₂, d₂ > , … < u_n, d_n > with a set of order-inducing variables u_i (i = 1, 2, … n) and a parameter γ such that γ ∈ (0, + ∞), denoted by $\begin{matrix} IG-DHFSHGM (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots, < u_{n}, d_{n} >) = \\ \frac{1}{γ} (\otimes_{j = 1}^{n} {(γ d_{σ (j)}^{ρ_{σ (j)} (τ, L)})}^{ρ_{j} (μ, N) / \sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}} (τ, L)}), \end{matrix}$ (26)

where σ (i) indicates a permutation on u_i (i = 1, 2, …, n) with u_σ(j) being the jth largest value u_i (i = 1, 2, …, n), ρ_j (μ, N) is the Shapley value with respect to (w.r.t.) the associated fuzzy measure μ on N ={ 1, 2, …, n } for the jth ordered position and ρ_{d
_σ(j)} (τ, L) is the Shapley value w.r.t. the fuzzy measure τ on L ={ d_j } (j = 1, 2, …, n) for d_j (j = 1, 2, …, n).

Remark 9. if μ and τ are both additive measure, then the IG-DHFSHGM operator degenerates to the induced generalized dual hesitant fuzzy hybrid geometric mean (IG-DHFHGM) operator

$\begin{matrix} IG-DHFHGM (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots < u_{n}, d_{n} >) \\ = \frac{1}{γ} (\otimes_{j = 1}^{n} {(γ d_{σ (j)}^{ω_{d_{σ (j)}}})}^{w_{j} / \sum_{j = 1}^{n} w_{j} ω_{d_{σ (j)}}}), \end{matrix}$ (27) where w_j = μ (j) and ω_{d
_j} = τ (d_j) for j = 1, 2, …, n.

Remark 10. If u_i = u_j for all i, j = 1, 2, …, n, with i ≠ j. Then the IG-DHFSHGM operator degenerates to the generalized dual hesitant fuzzy Shapley hybrid ordered geometric mean (G-DHFSHOGM) operator $\begin{matrix} G-DHFSHOGM (d_{1}, d_{2}, \dots, d_{n}) \\ = \frac{1}{γ} (\otimes_{j = 1}^{n} {(γ d_{σ (j)}^{ρ_{σ (j)} (τ, L)})}^{ρ_{j} (μ, N) / \sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}} (τ, L)}), \end{matrix}$ (28)

where $d_{σ (j)}^{ρ_{σ (j)} (τ, L)}$ is the jth largest value of the Shapley weighted DHEs $d_{i}^{ρ_{d_{i}} (τ, L)}$ (i = 1, 2, … n).

Remark 11. If γ = 1, then the IG-DHFSHGM operator degenerates to the induced dual hesitant fuzzy Shapley hybrid geometric mean (I-DHFSHGM) operator $\begin{matrix} I-DHFSHGM (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots < u_{n}, d_{n} >) \\ = \otimes_{j = 1}^{n} {(d_{σ (j)}^{ρ_{σ (j)} (τ, L)})}^{ρ_{j} (μ, N) / \sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}} (τ, L)} . \end{matrix}$ (29)

Remark 12. If γ = 2, then the IG-DHFSHGM operator degenerates to the induced dual hesitant fuzzy Shapley hybrid quadratic geometric mean(I-DHFSHQGM) operator $\begin{matrix} I-DHFSHQGM (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots < u_{n}, d_{n} >) \\ = \frac{1}{2} (\otimes_{j = 1}^{n} {(2 d_{σ (j)}^{ρ_{σ (j)} (τ, L)})}^{ρ_{j} (μ, N) / \sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}} (τ, L)}) . \end{matrix}$ (30)

Remark 13. If μ (j) = 1/ - n for each j = 1, 2, … n, then the IG-DHFSHGM operator degenerates to the induced generalized dual hesitant fuzzy Shapley geometric mean (IG-DHFSGM) operator

$\begin{matrix} I - DHFSGM \\ (< u_{1}, d_{1} >, < u_{2}, d_{2} >, \dots < u_{n}, d_{n} >) \\ = \frac{1}{2} (\otimes_{j = 1}^{n} 2 d_{σ (j)}^{ρ_{σ (j)} (τ, L)}) . \end{matrix}$ (31)

Theorem 2.Let <u₁, d₁ (h₁, g₁)> , < u₂, d₂ (h₂, g₂) > , …, < u_n, d_n (h_n, g_n) > be a set of two tuples with a set of order-inducing variablesu_i (i = 1, 2, …, n) andd_i (i = 1, 2, …, n) being a collection of DHFEs in D, letμbe a fuzzy measure on the ordered setN = {1, 2, …, n}, and letτbe the associated fuzzy measure onL = { d_i } _{i∈{1,2,…,n}}. Then, their aggregated value using the IG-DHFSHGM operator is also a DHFE, denoted by

$\begin{matrix} IG - DHFSHGM (\begin{matrix} < u_{1}, d_{1} (h_{1}, g_{1}) >, \\ < u_{2}, d_{2} (h_{2}, g_{2}) >, \\ \dots, < u_{n}, d_{n} (h_{n}, g_{n}) > \end{matrix}) \\ = ⋃_{r_{σ (1)} \in h_{σ (1)}, η_{σ (1)} \in g_{σ (1), σ (2)} \in h_{σ (2),} η_{σ (2)} \in g_{σ (2), \dots, σ (n)} \in h_{σ (n)}, η_{σ (n)} \in g_{σ (n)}} \\ {{1 - {(1 - \prod_{j = 1}^{n} {(1 - {(1 - r_{σ (j)}^{ρ_{d_{σ (j)}} (τ, L)})}^{γ})}^{ρ_{j} (μ, N) / - \sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}} (τ, L)})}^{1 / γ}}, \\ {\prod_{j = 1}^{n} {(1 - {(1 - {(1 - {(1 - η_{σ (j)})}^{ρ_{d_{σ (j)}} (τ, L)})}^{γ})}^{ρ_{j} (μ, N) / - \sum_{j = 1}^{n} ρ_{j} (μ, N) ρ_{d_{σ (j)}} (τ, L)})}^{1 / γ}}}, \end{matrix}$ (32) where the notations as shown in Definition 10.

The proof of Theorem 2 is similar to Theorem 1, and it is omitted here.

From Theorems 1 and 2, it is not difficult to know that IG-DHFSHWA and IG-DHFSHGM operators satisfy commutativity, monotonicity and boundary. Here, we no longer discuss them in detail.

4 Models for the optimal fuzzy measures with dual hesitant fuzzy information

For a decision-making problem, Let A = {A₁, A₂, …, A_m} be a finite set of m alternatives and C = {C₁, C₂, …, C_n} be a set of n attributes. Suppose that all values assigned to alternatives with respect to attributes are expressed by a dual hesitant fuzzy decision matrix denoted by D = (d_ij) _m×n, where elements d_ij = {h_ij, g_ij} are DHFEs provided for the ratings of the alternatives A_i (i = 1, 2, …, m) with respect to the attributes C_j (j = 1, 2, …, n), with h_ij = ⋃ _{r_ij∈h_ij} {r_ij} and g_ij = ⋃ _{η_ij∈g_ij} {η_ij}. If the weight information about attributes and their ordered positions are already known, then we can use relative aggregation operators to calculate the comprehensive attribute variables. However, in real decision making, due to time pressure, lack of knowledge or decision maker’s limited expertise about the problem domain, the weight information about attributes and their ordered positions are partly known or incompletely known, we should first determine the weights of attributes and their ordered positions.

The similarity measure analysis (SMA) method [52] as an important multi-attribute decision making method has been researched many scholars [52 –55]. To define the similarity relation coefficient for DHFEs, we first introduce the following distance measures on DHFEs.

Definition 11. Let d₁ = (h₁, g₁) and d₂ = (h₂, g₂) be any two DHFEs in Ω, then the distance between d₁ and d₂ is defined by $\begin{matrix} d (d_{1}, d_{2}) = \frac{1}{# h 1 + # h 2 + # g 1 + # g 2} \\ {(\begin{matrix} \sum_{r_{1} \in h_{1}} min_{r_{2} \in h_{2}} | r_{1} - r_{2} | + \sum_{r_{2} \in h_{2}} min_{r_{1} \in h_{1}} | r_{2} - r_{1} | \\ + \sum_{ϑ_{1} \in g_{1}} min_{ϑ_{2} \in g_{2}} | ϑ_{1} - ϑ_{2} | + \sum_{ϑ_{2} \in g_{2}} min_{ϑ_{1} \in g_{1}} | ϑ_{2} - ϑ_{1} | \end{matrix})}, \end{matrix}$ (33)

where # h₁, # h₂ and # g₁, # g₂ respectively denote the numbers of elements in h₁, h₂ and g₁, g₂.

Theorem 3.Letd₁andd₂be any two DHFEs inΩ, then the distance betweend₁andd₂defined Equation (33) satisfies the following properties:

(P1) 0 ≤ d (d₁, d₂) ≤ 1;

(P2) d (d₁, d₂) = 0 if and only if d₁ = d₂;

(P3) d (d₁, d₂) = d (d₂, d₁).

For the dual hesitant fuzzy decision matrix D = (d_ij) _m×n, let

$H^{+} = {H_{i 1}^{+}, H_{i 2}^{+}, \dots, H_{in}^{+}} (i = 1, 2, \dots, m)$ (34) $H^{-} = {H_{i 1}^{-}, H_{i 2}^{-}, \dots, H_{in}^{-}} (i = 1, 2, \dots, m),$ (35) where $H_{ij}^{+} = H_{σ (m) j}$ and $H_{ij}^{-} = H_{σ (1) j}$ for each j = 1, 2, …, n, and σ (i) indicates a permutation on M ={ 1, 2, …, m } such that H_σ(1)j ≤ H_σ(2)j ≤ H_σ(m)j. Then, the similarity of d_ij to $H_{ij}^{+}$ is defined by Equation (36)

$ζ_{ij}^{+} = 1 - \frac{d (d_{ij}, H_{ij}^{+})}{\sum_{j = 1}^{n} d (d_{ij}, H_{ij}^{+})},$ (36) where $d (d_{ij}, H_{ij}^{+})$ denotes the distance between the DHFEs d_ij to $H_{ij}^{+}$ defined as in Equation (33).

Similarly, the similarity of d_ij to $H_{ij}^{-}$ is calculated as follows: $ζ_{ij}^{-} = 1 - \frac{d (d_{ij}, H_{ij}^{-})}{\sum_{j = 1}^{n} d (d_{ij}, H_{ij}^{-})},$ (37) where $d (d_{ij}, H_{ij}^{-})$ denotes the distance between the DHFEs d_ij to $H_{ij}^{-}$ defined as in Equation (33) (i = 1, 2, …, m ; j = 1, 2, …, n).

If the weight information of the attributes is partly known, we construct the following model for the optimal fuzzy measure τ on the attributes set C w.r.t. the alternative A_i (i = 1, 2, …, m). $(M - 1) {\begin{matrix} max \sum_{j = 1}^{n} \frac{ζ_{ij}^{+}}{ζ_{ij}^{-} + ζ_{ij}^{+}} ρ_{c_{j}} (τ, C) \\ s . t . τ (C) = 1, τ (K) \leq τ (P), \\ \forall K, P \in Cs . t . K \subseteq P, \\ τ (c_{j}) \in U_{c_{j}}, τ (c_{j}) \geq 0, j = 1, 2, \dots n . \end{matrix}$ (38) where ρ_{c
_j} (τ, C) is the Shapley value of the attributes with respect to the associated fuzzy measure τ and U_{c
_j} is the known weight information (j = 1, 2, … n).

Since all alternatives are non inferior, we further construct the following model for the optimal fuzzy measure τ on the attributes set C, the model (M - 1) are transformed into the following form: $(M - 2) {\begin{matrix} max \sum_{i = 1}^{m} \sum_{j = 1}^{n} \frac{ζ_{ij}^{+}}{ζ_{ij}^{-} + ζ_{ij}^{+}} ρ_{c_{j}} (τ, C) \\ s . t . τ (C) = 1, τ (K) \leq τ (P), \\ \forall K, P \in Cs . t . K \subseteq P, \\ τ (c_{j}) \in U_{c_{j}}, τ (c_{j}) \geq 0, j = 1, 2, \dots n . \end{matrix}$ (39)

Now, we consider the optimal fuzzy measure on the ordered set N.

For the dual hesitant fuzzy decision matrix D = (d_ij) _m×n, let $\bar{D} = {{\bar{d}}_{1}, {\bar{d}}_{2}, \dots, {\bar{d}}_{n}}$ , where ${\bar{d}}_{j} = (1 / - m) \oplus_{i = 1}^{m} d_{ij}$ for each j = 1, 2, …, n.

The similarity of d_ij to ${\bar{d}}_{j}$ is calculated by ${\bar{ζ}}_{ij} = 1 - \frac{d (d_{ij}, {\bar{d}}_{ij})}{\sum_{j = 1}^{n} d (d_{ij}, {\bar{d}}_{ij})} .$ (40)

For each i = 1, 2, …, m, reorder ${\bar{ζ}}_{ij}$ (j = 1, 2, …, n) such that ${\bar{ζ}}_{i (j)}$ being the jth largest value of ${\bar{ζ}}_{ij} (j = 1, 2, \dots, n)$ . If the ordered position weight information is partly known, we construct the following model for the optimal fuzzy measure μ on the ordered set N w.r.t. the alternative A_i (i = 1, 2, …, m). $(M - 3) {\begin{matrix} min \sum_{j = 1}^{n} {\bar{ζ}}_{i (j)} ρ_{j} (μ, N) \\ s . t . μ (N) = 1, μ (K) \leq μ (P), \\ \forall K, P \in Cs . t . K \subseteq P, \\ μ (j) \in U_{j}, μ (j) \geq 0, j = 1, 2, \dots n, \end{matrix}$ (41) where ρ_j (μ, N) is the Shapley value of the attributes with respect to the associated fuzzy measure μ and U_j is the known weight information (j = 1, 2, … n). For different alternatives, there is not any difference between fuzzy measures on the ordered set N. We further construct the following model for the optimal fuzzy measure μ on the ordered set N. $(M - 4) {\begin{matrix} min \sum_{i = 1}^{m} \sum_{j = 1}^{n} {\bar{ζ}}_{i (j)} ρ_{j} (μ, N) \\ s . t . μ (N) = 1, μ (K) \leq μ (P), \\ \forall K, P \in Cs . t . K \subseteq P, \\ μ (j) \in U_{j}, μ (j) \geq 0, j = 1, 2, \dots n . \end{matrix}$ (42)

Remark 14. In models (39) and (42), we use the elements’ Shapley values as their weights, which globally consider their importance as well as overall reflect their correlations on them. If there is not any interaction among attributes and among their ordered positions, then the models (39) and (42) respectively degenerates to models for optimal additive weight vectors on the attributes set and on the ordered set.

5 An approach to dual hesitant fuzzy multi-attributes decision making

Step 1. There exist m alternatives A = {A₁, A₂, …, A_m} with respect to n attributes C = {C₁, C₂, …, C_n} to be evaluated to form dual hesitant fuzzy decision matrix D = (d_ij) _m×n, where d_ij is a DHFE.

Step 2. Assume all the attributes C_j (j = 1, 2, …, n) are of the same type. In such cases, the attribute values of cost type may be transformed into attribute values of the benefit types.

Step 3. Utilize model (39) to solve the optimal fuzzy τ on the attribute set C, and calculate their Shapley values.

Step 4. Utilize model (42) to solve the optimal fuzzy μ on the attribute set N, and calculate their Shapley values.

Step 5. Let $u_{j} = {\bar{ζ}}_{ij}$ (j = 1, 2, …, n) for each (i = 1, 2, …, m), utilize the induced generalized dual hesitant fuzzy Shapley hybrid weight averaging (IG-DHFSHWA) operator or induced generalized dual hesitant fuzzy Shapley hybrid geometric mean (IG-DHFSHGM) operator to calculate the comprehensive DHFEs d_i (i = 1, 2, …, m) of the alternative A_i (i = 1, 2, …, m).

Step 6. According to the comprehensive DHFEs d_i (i = 1, 2, …, m), calculate the score value S (d_i) and the averaging deviation value D (d_i). Then, to rank the comprehensive DHFEs d_i (i = 1, 2, …, m), and select the best alternative(s).

Step 7. End.

6 Numerical example

A high-technology manufacturing center in an automaker desires to select a suitable green material supplier to purchase the key components of new products. After preliminary screening, five candidates A_i (i = 1,2,3,4,5) remain for further evaluation. Five benefit criteria are considered: (1) Green supplier chain (GSC) organization (C₁); (2) GSC financial performance (C₂); (3) GSC service quality (C₃); (4) GSC technology (C₄); (5) GSC competencies (C₅). To avoid influence each other, the decision makers are required to evaluate the five possible alternatives A_i (i = 1, 2, 3, 4, 5) under the above five attributes in anonymity and construct the following dual hesitant fuzzy decision matrix D = (d_ij) _5×5 is presented in Table 1, where d_ij (i = 1, 2, 3, 4, 5, j = 1, 2, 3, 4, 5) are in the form of DHFEs.

Due to the time restraints and the limited expertise of the experts about the researches field, the weight information of the attributes is only partly known and respectively given as: $\begin{matrix} U_{C_{1}} & = & [0.3, 0.5], U_{C_{2}} = [0.35, 0.5], \\ U_{C_{3}} & = & [0.2, 0.35], U_{C_{4}} = [0.25, 0.4], \\ U_{C_{5}} & = & [0.2, 0.4] . \end{matrix}$

Further, the importance of the ordered positions is respectively defined by $\begin{matrix} U_{1} & = & [0.15, 0.3], U_{2} = [0.2, 0.3], \\ U_{3} & = & [0.3, 0.5], U_{4} = [0.3, 0.5], \\ U_{5} & = & [0.25, 0.4] . \end{matrix}$

To effectively solve the problem, we can utilize the above proposed procedure to get the most desirable alternative(s).

Step 1. Since all the attributes C_j (j = 1, 2, …, 5) are of the benefit, the attribute values do not need normalization. Therefore, D = D′.

Step 2. Identify the dual hesitant fuzzy positive ideal solutions H⁺ and dual hesitant fuzzy negative solutions H^– in the dual hesitant fuzzy decision matrix D by Equations (34) and (35), respectively. $\begin{matrix} H^{+} \\ = {\begin{matrix} {{0.8}, {0.1}}, {{0.6, 0.7, 0.8}, {0.2}}, \\ {{0.5, 0.6}, {0.2}}, {{0.5, 0.6, 0.7}, {0.3}}, \\ {{0.9}, {0.1}} \end{matrix}}, \\ H^{-} \\ = {\begin{matrix} {{0.2, 0.3}, {0.6}}, {{0.3}, {0.6, 0.7}}, \\ {{0.3}, {0.5, 0.6, 0.7}}, {{0.3}, {0.5, 0.6, 0.7}}, \\ {{0.5, 0.6}, {0.4}} \end{matrix}} . \end{matrix}$

Step 3. Calculate the similarity of each alternative form the positive ideal solutions H⁺ and H^– by Equations (36) and (37) respectively, as follows: $\begin{matrix} ζ^{+} & = & (\begin{matrix} 0.5750 & 0.9080 & 0.8229 & 0.9066 & 0.7875 \\ 1.0000 & 0.7183 & 0.4551 & 0.8266 & 1.0000 \\ 0.7694 & 0.6706 & 0.8235 & 1.0000 & 0.7365 \\ 0.8554 & 1.0000 & 0.6867 & 0.6024 & 0.85554 \\ 0.8427 & 0.6348 & 1.0000 & 0.7472 & 0.7753 \end{matrix}), \\ ζ^{-} & = & (\begin{matrix} 1.0000 & 0.5714 & 0.8286 & 0.8000 & 0.8000 \\ 0.5439 & 0.8371 & 1.0000 & 0.8997 & 0.7193 \\ 0.6333 & 1.0000 & 0.8750 & 0.4500 & 1.0000 \\ 0.6573 & 0.6737 & 0.8485 & 1.0000 & 0.8205 \\ 0.6348 & 0.9478 & 0.6174 & 0.9391 & 0.8609 \end{matrix}) . \end{matrix}$

From the similarity matrices ζ⁺ and ζ^-, we obtain the following relative similarity matrix $\begin{matrix} \frac{ζ^{+}}{ζ^{+} + ζ^{-}} \\ = (\begin{matrix} 0.3651 & 0.6138 & 0.4983 & 0.5312 & 0.4961 \\ 0.6477 & 0.4618 & 0.3128 & 0.4788 & 0.5816 \\ 0.5485 & 0.4014 & 0.4848 & 0.6897 & 0.4241 \\ 0.5655 & 0.5975 & 0.4473 & 0.3759 & 0.5104 \\ 0.5704 & 0.4011 & 0.6183 & 0.4431 & 0.4738 \end{matrix}) . \end{matrix}$

According to model (39), the optimal fuzzy measure τ on the attributes set C are obtained as shown in Table 2.

According to the Table 2, it derives the following Shapley values $\begin{matrix} ρ_{c_{1}} (τ, C) & = & 0.6525, ρ_{c_{2}} (τ, C) = 0.1275, \\ ρ_{c_{3}} (τ, C) & = & 0.1025, ρ_{c_{4}} (τ, C) = 0.0775, \\ ρ_{c_{5}} (τ, C) & = & 0.0775 \end{matrix}$

Step 4. Form Equation (40), it gets the following similarity matrix. $\bar{ζ} = (\begin{matrix} 0.9958 & 0.0060 & 0.9993 & 0.9997 & 0.9992 \\ 0.9971 & 0.0062 & 0.9993 & 0.9994 & 0.9979 \\ 0.9993 & 0.0041 & 0.9996 & 0.9988 & 0.9981 \\ 0.9988 & 0.0043 & 0.9991 & 0.9985 & 0.9993 \\ 0.9994 & 0.0048 & 0.9977 & 0.9990 & 0.9990 \end{matrix}) .$

According to model (42), the optimal fuzzy measure μ on the ordered set N are obtained as shown in Table 3.

According to the Table 3, it derives the following Shapley values $\begin{matrix} ρ_{1} (μ, N) & = & 0.06, ρ_{2} (μ, N) = 0.0725, \\ ρ_{3} (μ, N) & = & 0.1058, ρ_{4} (μ, N) = 0.1058, \\ ρ_{5} (μ, N) & = & 0.6558 . \end{matrix}$

Step 5. Utilize the IG-DHFSHWA operator to calculate the comprehensive DHFEs (i = 1 -5) for the green supplier A_i (i = 1 -5).

Step 6. According to the comprehensive DHFEs d_i (i = 1, …, m), calculate the score value S (d_i) and the averaging deviation value D (d_i). Then rank the comprehensive DHFEs d_i (i = 1, …, m), and select the best alternative(s): $\begin{matrix} S (d_{1}) & = & - 0.0304, S (d_{2}) = 0.4276, \\ S (d_{3}) & = & - 0.1120, S (d_{4}) = 0.4667, \\ S (d_{5}) & = & 0.0487 . \end{matrix}$

Therefore, the most desirable alternatives is A₄.

In the above example, we only use theIG-DHFSHWA operator γ = 1 to obtain the best choice. The corresponding score values and the ranking order of alternatives based on the IG-DHFSHWA operator with different values of parameter γ was shown in Table 4.

In order to illustrate how the different parameter γ plays a role in the aggregation operator, we use the different values of γ ∈ (0, 10], which are given by the decision-makers. The changing trends of global preference score function S (d_i) (i = 1, 2, 3, 4) with respect to γ in IG-DHFSHWA are shown in Fig. 1.

Figure 1 illustrates the scores of the alternatives obtained by the IG-DHFSHWA operator as γ assigned different values.

Furthermore, from Fig. 1, we can find that,

when γ ∈ (0, 0.8], the ranking of the five alternatives is A₄ ≻ A₂ ≻ A₁ ≻ A₅ ≻ A₃.

when γ ∈ (0.8, 1.2], the ranking of the five alternatives is A₄ ≻ A₂ ≻ A₅ ≻ A₁ ≻ A₃.

when γ ∈ (1.2, 1.3], the ranking of the five alternatives is A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃.

when γ ∈ (1.3, 10], the ranking of the five alternatives is A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁.

Parallel to IG-DHFSHWA operator, the comprehensive DHFEs (i = 1 -5) for the green supplier A_i (i = 1 -5) based on the IG-DHFSHGM operator with different values of parameter γ was shown in Table 5.

Different rankings of the alternatives can be obtained with different values of γ ∈ (0, 10] by IG-DHFSHGM operator which was shown inFig. 2. From Fig. 2, we can find that,

when γ ∈ (0, 1.67], the ranking of the five alternatives is A₄ ≻ A₂ ≻ A₅ ≻ A₃ ≻ A₁.

when γ ∈ (1.67, 2.6], the ranking of the five alternatives is A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁.

when γ ∈ (2.6, 2.637], the ranking of the five alternatives is A₂ ≻ A₅ ≻ A₄ ≻ A₃ ≻ A₁.

when γ ∈ (2.637, 10], the ranking of the five alternatives is A₂ ≻ A₅ ≻ A₃ ≻ A₄ ≻ A₁.

In order to show the merit of the proposed method, we utilized some existing methods to solve this numerical example. The decision making method introduced by Wang et al. [35] has distinct differences form the method presented in Section 6. The following main steps are involved based on the method proposed by Wang et al. [35].

In the following, we use the generalized dual hesitant fuzzy Choquet ordered average (GDHFCOA) operator and the generalized dual hesitant fuzzy Choquet ordered geometric (GDHFCOG) operator to aggregate the dual hesitant fuzzy information. The aggregated dual hesitant fuzzy elements can obtained when a determined value are set to parameter γ. Tables 4 and 5 show the score function of the aggregated dual hesitant fuzzy elements and ranking of the alternatives are also presented in Tables 6 and 7.

Different rankings of the alternatives can be obtained with different values of γ ∈ (0, 10] by GDHFCOA operator which was shown in Fig. 3.

From Fig. 3, we can find that the score values of the five alternatives obtained by the GDHFCOA operator become larger with increase of the parameter γ. Decision-makers can choose the values of parameter γ in accordance with their preferences. There is not any influence on the final rankings of the alternatives for parameter γ ∈ (0, 10].

Further, we can find that γ ∈ (0.10], the ranking of the five alternatives is A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁.

Different rankings of the alternatives can be obtained with different values of γ ∈ (0, 10] by GDHFCOG operator which was shown in Fig. 4.

It is easy to see from Fig. 4 that the score values obtained by the GDHFCOG operator become smaller with increase of the parameter γ.

Further, we can find that,

when γ ∈ (0, 0.7], the ranking of the five alternatives is A₂ ≻ A₄ ≻ A₅ ≻ A₃ ≻ A₁.

when γ ∈ (0.7, 10], the ranking of the five alternatives is A₂ ≻ A₄ ≻ A₅ ≻ A₁ ≻ A₃.

The comprehensive comparison has been summarized from the following three aspects.

The ranking results of all feasible alternatives are different as γ changes. The reason may be yielded by using the different aggregation operators, and thus, the decision makers can properly select the desirable alternative according to his interest and the actual needs.

The existing GDHFCOA and GDHFCOG operators only reflect the interaction between two adjacent coalitions and give the importance of elements or that of ordered positions, but it cannot both consider these two aspects, while the IG-DHFSHWA and IG-DHFSHGM operators not only globally consider the importance of elements and their ordered positions, but also overall reflect the interaction among them and among their ordered positions, which could be used and explained reasonably in more case.

Dual hesitant fuzzy information fusion method is a useful research topic of the fuzzy decision science. In this paper, we propose the new dual hesitant fuzzy information operators, which furnish a beautiful complement to the accepted research results on dual hesitant fuzzy set theory.

7 Conclusion

In this paper, we investigate the multiple attribute decision making method in which the attribute values take the form of dual hesitant fuzzy elements with incomplete weight information and interactive conditions. In order to get the comprehensive attribute values, two new induced generalized dual hesitant fuzzy aggregation operators called the induced generalized dual hesitant fuzzy Shapley hybrid weighted averaging (IG-DHFSHWA) operator and the induced generalized dual hesitant fuzzy Shapley hybrid geometric mean (IG-DHFSHGM) operator are defined, which do not only consider the importance of the elements and their ordered positions but also reflect their interactions. In many practical decision-making problems, the weight vectors about attribute set and the ordered set is incomplete or partly known. To obtain their optimal weights vectors, the corresponding models are established by the using the similarity measure analysis (SMA) method and the Shapley function.

Besides the hybrid aggregation operator, many researchers investigated some other kinds of interesting aggregation operators. For example, Torra [56] defined the weighted ordered weighted- average (WOWA) operator based on a monotonic function and the advantages of the OWA operator and the ones of the weighted mean. Furthermore, Yager [57] developed an approach to the inclusion of importances in the OWA operator aggregation technique, and Merigó [58] defined the weights using weighted average of the weights of the elements and that of the ordered positions and presented the induced ordered weighted averaging-weighted average (IOWAWA) operator. It is one of our future research works to consider these operators in the setting of dual hesitant fuzzy environment based on fuzzy measures.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61175122, 71071018, 71201089), the Fundamental Research Funds for the Central Universities (No. JBK150502, 14XJC630010), Special Innovation Project of Guangdong University of Foreign Studies (No. 15T21), Social Science, Natural Science, Soft Science, and Science Innovation Project of Guangdong Province (No. GD12XGL14, 2014A030313575, 2015A070704051, 14G41, 2013KJCX0072).

References

Zadeh

L.A.

, Fuzzy sets, Information and Control8(3) (1965), 338–353.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems20(1) (1986), 87–96.

Atanassov

and Gargov

, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems31(3) (1989), 343–349.

Turksen

I.B.

, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems20(2) (1986), 191–210.

Mamdani

E.H.

and Assilian

, An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man-Machine Studies7(1) (1975), 1–13.

Karnik

N.N.

, Mendel

J.M.

and Liang

, Type-2 fuzzy logic systems, IEEE Transactions on Fuzzy Systems7(6) (1999), 643–658.

Mendel

J.M.

and John

R.B.

, Type-2 fuzzy sets made simple, IEEE Transactions on Fuzzy Systems10(2) (2002), 117–127.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems25(6) (2010), 529–539.

Miyamoto

, Multisets and fuzzy multisets, in: Soft Computing and Human-Centered Machines, Springer, Berlin, Germany, 2000, pp. 9–33.

10.

Torra

and Narukawa

, On hesitant fuzzy sets and decision, in: The 18th International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378–1382.

11.

Xia

M.M.

and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning52(3) (2011), 395–407.

12.

Wei

G.W.

, Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowledge-Based Systems31 (2012), 176–182.

13.

Z.S.

and Xia

M.M.

, Distance and similarity measures for hesitant fuzzy sets, Information Sciences181(11) (2011), 2128–2138.

14.

Wei

G.W.

, Zhao

X.F.

, Wang

H.J.

and Lin

, Hesitant fuzzy Choquet integral aggregation operators and their applications to multiple attribute decision making, Information-An International Interdisciplinary Journal15(2) (2012), 441–448.

15.

Wei

G.W.

and Zhang

, A multiple criteria hesitant fuzzy decision making with Shapley value-based VIKOR method, Journal of Intelligent & Fuzzy Systems26(2) (2014), 1065–1075.

16.

Chen

, Xu

Z.S.

and Xia

M.M.

, Interval-valued hesitant preference relations and their applications to group decision making, Knowledge-Based Systems37 (2013), 528–540.

17.

Chen

, Xu

Z.S.

and Xia

M.M.

, Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis, Applied Mathematical Modelling37(4) (2013), 2197–2211.

18.

Wei

G.W.

, Zhao

X.F.

and Lin

, Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making, Knowledge-Based Systems46 (2013), 43–53.

19.

Herrera

and Herrera-Viedma

, Linguistic decision analysis: Steps for solving decision problems under linguistic information, Fuzzy Sets and Systems115(1) (2000), 67–82.

20.

Rodriguez

R.M.

, Martinez

and Herrera

, Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Fuzzy Systems20(1) (2012), 109–119.

21.

Lin

, Zhao

X.F.

and Wei

G.W.

, Models for selecting an ERP system with hesitant fuzzy linguistic information, Journal of Intelligent & Fuzzy Systems26(5) (2014), 2155–2165.

22.

Meng

F.Y.

, Chen

X.H.

and Zhang

, Multi-attribute decision analysis under a linguistic hesitant fuzzy environment, Information Sciences267 (2014), 287–305.

23.

Zhu

, Xu

Z.S.

and Xia

M.M.

, Dual Hesitant Fuzzy Sets, Journal of Applied Mathematics2012 (2012), 1–13.

24.

Zadeh

L.A.

, A fuzzy-algorithmic approach to the definition of complex or imprecise concepts, International Journal of Man-Machine Studies8(3) (1976), 249–291.

25.

Yager

R.R.

, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Transactions on Systems, Man and Cybernetics18(1) (1988), 183–190.

26.

Wang

H.J.

, Zhao

X.F.

and Wei

G.W.

, Dual hesitant fuzzy aggregation operators in multiple attribute decision making, Journal of Intelligent & Fuzzy Systems26(5) (2014), 2281–2290.

27.

Yang

S.H.

and Ju

Y.B.

, Dual hesitant fuzzy linguistic aggregation operators and their applications to multi-attribute decision making, Journal of Intelligent & Fuzzy Systems27(4) (2014), 1935–1947.

28.

Tan

C.Q.

, A multi-criteria interval-valued intuitionistic fuzzy group decision making with Choquet integral-based TOPSIS, Expert Systems with Applications38(4) (2011), 3023–3033.

29.

Meng

F.Y.

, Tan

C.Q.

and Zhang

, The induced generalized interval-valued intuitionistic fuzzy hybrid Shapley averaging operator and its application in decision making, Knowledge-Based Systems42 (2013), 9–19.

30.

Z.S.

, Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences180(5) (2010), 726–736.

31.

J.Z.

, Chen

, Nie

C.P.

and Zhang

, Intuitionistic fuzzy-valued Choquet integral and its application in multicriteria decision making, Information Sciences222 (2013), 509–527.

32.

Meng

F.Y.

and Zhang

, Induced continuous Choquet integral operators and their application to group decision making, Computers & Industrial Engineering68 (2014), 42–53.

33.

Meng

F.Y.

and Chen

X.H.

, An approach to uncertain linguistic multi-attribute group decision making based on interactive index, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems23(3) (2015), 319–344.

34.

Meng

F.Y.

, Tan

C.Q.

and Chen

X.H.

, An approach to Atanassov’s interval-valued intuitionistic fuzzy multi-attribute decision making based on prospect theory, International Journal of Computational Intelligence Systems8(3) (2015), 591–605.

35.

Wang

C.Y.

, Li

Q.G.

and Zhou

X.Q.

, Multiple attribute decision making based on generalized aggregation operators under dual hesitant fuzzy environment, Journal of Applied Mathematics2014 (2014), 1–12.

36.

Y.B.

, Yang

S.H.

and Liu

X.Y.

, Some new dual hesitant fuzzy aggregation operators based on Choquet integral and their applications to multiple attribute decision making, Journal of Intelligent & Fuzzy Systems27(6) (2014), 2857–2868.

37.

Meng

F.Y.

, Zhang

and Cheng

, Approaches to multiple-criteria group decision making based on interval-valued intuitionistic fuzzy Choquet integral with respect to the generalized λ-Shapley index, Knowledge-Based Systems37 (2013), 237–249.

38.

Meng

F.Y.

and Chen

X.H.

, An approach to interval-valued hesitant fuzzy multi-attribute decision making with incomplete weight information based on hybrid Shapley operators, Informatica25(4) (2014), 617–642.

39.

Meng

F.Y.

, Tan

C.Q.

and Zhang

, Some interval-valued intuitionistic uncertain linguistic hybrid Shapley operators, Journal of Systems Engineering and Electronic25(3) (2014), 452–463.

40.

Meng

F.Y.

and Chen

X.H.

, A hesitant fuzzy linguistic multi-granularity decision making model based on distance measures, Journal of Intelligent & Fuzzy Systems28(4) (2015), 1519–1531.

41.

D.J.

, Some generalized dual hesitant fuzzy geometric aggregation operators and applications, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems22(3) (2014), 367–384.

42.

Sugeno

, Theory of fuzzy integral and its applications, Doctorial Dissertation, Tokyo Institute of Technology, 1974.

43.

Shapley

L.S.

, A value for n-person games, Princeton: Princeton University Press, 1953.

44.

Marichal

J.L.

, The influence of variables on pseudo-Boolean functions with applications to game theory and multicriteria decision making, Discrete Applied Mathematics107(1) (2000), 139–164.

45.

Merigo

J.M.

, Peris-Ortiz

and Palacios-Marques

, Entrepreneurial fuzzy group decision-making under complex environments, Journal of Intelligent & Fuzzy Systems27(2) (2014), 901–912.

46.

Merigó

J.M.

and Casanovas

, Fuzzy generalized hybrid aggregation operators and its application in fuzzy decision making, International Journal of Fuzzy Systems12(1) (2010), 15–24.

47.

Yager

R.R.

and Filev

D.P.

, Induced ordered weighted averaging operators, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics29(2) (1999), 141–150.

48.

Merigó

J.M.

and Gil-Lafuente

A.M.

, The induced generalized OWA operator, Information Sciences179(6) (2009), 729–741.

49.

Merigó

J.M.

and Gil-Lafuente

A.M.

, Fuzzy induced generalized aggregation operators and its application in multi-person decision making, Expert Systems with Applications38(8) (2011), 9761–9772.

50.

Y.J.

and Wang

H.M.

, The induced generalized aggregation operators for intuitionistic fuzzy sets and their application in group decision making, Applied Soft Computing12(3) (2012), 1168–1179.

51.

Z.X.

, Xia

G.P.

, Chen

M.Y.

and Wang

, Induced generalized intuitionistic fuzzy OWA operator for multi-attribute group decision making, Expert Systems with Applications39(2) (2012), 1902–1910.

52.

Y.H.

, Olson

D.L.

and Qin

, Similarity measures between intuitionistic fuzzy (vague) sets: A comparative analysis, Pattern Recognition Letters28(2) (2007), 278–285.

53.

Wei

G.W.

, Grey relational analysis method for 2-tuple linguistic multiple attribute group decision making with incomplete weight information, Expert Systems with Applications38(5) (2011), 4824–4828.

54.

Kim

S.H.

, Choi

S.H.

and Kim

J.K.

, An interactive procedure for multiple attribute group decision making with incomplete information: Range-based approach, European Journal of Operational Research118(1) (1999), 139–152.

55.

Zhang

, Jin

and Liu

, A grey relational projection method for multi-attribute decision making based on intuitionistic trapezoidal fuzzy number, Applied Mathematical Modelling37(5) (2013), 3467–3477.

56.

Torra

, The weighted OWA operator, International Journal of Intelligent Systems12(2) (1997), 153–166.

57.

Yager

R.R.

, Including importances in OWA aggregations using fuzzy systems modeling, IEEE Transactions on Fuzzy Systems6(2) (1998), 286–294.

58.

Merigó

J.M.

, A unified model between the weighted average and the induced OWA operator, Expert Systems with Applications38(9) (2011), 11560–11572.